`int(sinsqrt(x))/(sqrt(x))dx=?`

Options

A. `2cossqrt(x)+C`
B. `-2cossqrt(x)+C`
C. `-(cossqrt(x))/(2)+C`
D. `(cossqrt(x))/(2)+C`

Explanation

To solve the integral \(\int \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx\), we will use a substitution method. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( t = \sqrt{x} \). Then, we can express \( x \) in terms of \( t \):
\[ x = t^2 \]
Now, we differentiate both sides to find \( dx \):
\[ dx = 2t \, dt \]

### Step 2: Rewrite the Integral
Now, we substitute \( \sqrt{x} \) and \( dx \) into the integral:
\[ \int \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx = \int \frac{\sin(t)}{t} \cdot 2t \, dt \]
This simplifies to:
\[ \int 2 \sin(t) \, dt \]

### Step 3: Integrate
Now we can integrate:
\[ \int 2 \sin(t) \, dt = -2 \cos(t) + C \]

### Step 4: Back Substitute
Now we need to substitute back \( t = \sqrt{x} \):
\[ -2 \cos(t) + C = -2 \cos(\sqrt{x}) + C \]

### Final Answer
Thus, the integral is:
\[ \int \frac{\sin(\sqrt{x})}{\sqrt{x}} \, dx = -2 \cos(\sqrt{x}) + C \]